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Math 341

__Side-Angle-Side on the Sphere__
Recall the following
fact about planar triangles:

(SAS)
If two triangles have the two sides equal to two sides respectively, and
have the angles contained by the equal straight lines equal, they will
also have the base equal to the base, the triangle will be equal to the
triangle, and the remaining angles will be equal to the remaining angles
respectively, namely those which the equal sides subtend.

This is Euclid’s
Proposition 4.

Decide whether or not SAS
is true on the sphere. If you think that it is true on the sphere,
then give a convincing argument that shows that it is true. If you
think that it isn’t true, explain why it is false, show how to modify the
statement to make it true on the sphere, and then prove your modified statement.
In either case, explain and justify any decisions that you had to make
in order to come to your conclusion. Try to write explanations that
would convince a reasonable skeptic. You will probably find it helpful
to thoroughly understand Euclid’s planar proof before trying to decide
what happens on the sphere.